Three-layer stratified exchange flows: hydraulically controlled transition to turbulence

Buoyancy-driven exchange flows in geophysical contexts often exhibit significant interfacial turbulence leading to a partially mixed intermediate layer between two counterflowing layers. In this paper we perform a three-layer hydraulic analysis of such flows, highlighting the dynamical importance of the middle mixed layer. Our analysis is based on the viscous, shallow water, Boussinesq equations and includes the effects of mixing as a non-hydrostatic pressure forcing. We demonstrate the superior predictive accuracy of three-layer hydraulics over the more classical two-layer approach by applying it to direct numerical simulation data in stratified inclined duct exchange flows where turbulence is controlled by a modest slope of the duct. The three-layer model predicts a region bounded by two control points in the middle of the duct, linked to the onset of instability and turbulence, whereas a two-layer model only predicts one control point. We show that the nonlinear characteristics of the three-layer model correspond to linear long waves perturbing a three-layer mean flow. We also provide the first evidence of long-wave resonance, as well as resonance between long and short waves, and their connection to turbulence. These results challenge current parameterisations for turbulent transport, which typically overlook long waves and internal hydraulics induced by streamwise variations of the flow.